SinglePhoton Imaging Spectroscopy

1
Single-Photon Imaging Spectroscopy
Imaging Spectroscopy meeting, 6th RHESSI
workshop, April 3-4, Paris-Meudon

• Kaspar Arzner1), Alex Zehnder1), Marina
  Battaglia2), and Paolo Grigis2)

1) Paul Scherrer Institut, 2) Institut of
Astronomy, ETHZ
2
Idea

• Apply an origin cut on the event list with
  respect to known sources.
• Segregate eventlist accorinf to the most likley
  origin create energy histograms of each source.
• Apply traditional spectrum inversion.

NB This procedure pre-assumes a spatial source
distribution, and estimates (spectra image)
than (spectra, image) !
3
Transmission Probabilities

• Assume there are two or more distinct sources
  which can be modeled by
• Bi(x,y) probability that a photon
  emitted by source i comes from direction
  (x,y).
• The probability that a photon emitted by source i
  triggers the event e (j,t,E) is
• pi (e) ? dx dy M(x,y,e) Bi(x,y)
• where M(x,y,e) is the grid transmission
  (modulation pattern).
• Assume that each e originates from exactly one
  source. Then,
• ?i (e) pi (e) / ?k pk(e)
• can be interpreted as the probability that e
  comes from source i.

j .... detector t .... time E ... Energy
4
Selection of events

• Use only events with large maxi ?i(e).
• ?maxi ?i(e)? depends weakly on energy but
  strongly on the (assumed) source size. (Small
  alternative sources are more frequently covered
  and excluded.)

• Thus the absolute intensity of each source is not
  accessible, but the spectral shape is.
• Trade-off between number N of counts and
  assignment reliabilities
• Take for each source those N events with largest
  maxi ?i(e). Typically, N 1000.
• The absolute normalization may be restored a
  posteriori from fwdfit.

Example with 2 sources (details in a minute!)
5
Spatial Source Models

• Gaussian Bi(x,y) with parameters (ri,wi,fi)
  representing (position,shape,orientation).
• Grid transmission at event e(j,t,E) 1st
  harmonic only,
• M(r,e) a0(e)a1(e)cos(k(t)?(r-P(t))?j(
  t))
• The energy-dependence of a0 and a1 is
  interpolated from a logarithmic lattice (spacing
  factor 1.3).
• Thus,
• pi(e) a0(e) a1(e)e-kTSik/2
  cos(k(t)?(ri-P(t))?j(t))
• where Si R(fi)Tdiag(wix,wiy)R(fi), and R(f)
  represents clockwise rotation by f.

6
Selection of source parameters

• By eye, from CLEAN components.
• From forward-fit solutions.
• In cases of doubt, consult the maximum likelihood
  
• Lmax maxai L(eaiBi)
• achievable by varying the source amplitudes
  ai

solid by-eye dashed fwdfit graysale CLEAN
normalization
IDLgt maximize_a,p,p_event,S0,dtime,dt,Nitmax,
a,logL,diagnosticsdiagnostics 0th iteration
logL/N_cnt-6.52269e00 cond(H)1.89e01 act/s
25619.28 25660.18 1th iteration
logL/N_cnt-6.42796e00 cond(H)1.89e01 act/s
23397.89 22750.66 2th iteration
logL/N_cnt-6.36731e00 cond(H)1.89e01 act/s
21741.40 20714.16 ... 49th iteration
logL/N_cnt-6.21572e00 cond(H)1.90e01 act/s
13701.13 12005.24 50th iteration
logL/N_cnt-6.21572e00 cond(H)1.90e01 act/s
13697.70 12001.85 ?MAN
IDLgt maximize_a,p,p_event,S0,dtime,dt,Nitmax,
a,logL,diagnosticsdiagnostics 0th iteration
logL/N_cnt-6.52569e00 cond(H)1.75e01 act/s
25607.07 25683.66 1th iteration
logL/N_cnt-6.43103e00 cond(H)1.75e01 act/s
22878.17 23283.37 2th iteration
logL/N_cnt-6.37040e00 cond(H)1.75e01 act/s
20942.11 21526.62 ... 49th iteration
logL/N_cnt-6.21881e00 cond(H)1.76e01 act/s
12418.19 13296.46 50th iteration
logL/N_cnt-6.21881e00 cond(H)1.76e01 act/s
12414.80 13293.03 ?FF
7
Error estimates
Form energy histograms nik , i 1..Nsrc , k
1..NE.

• Statistical errors ?nik
• Systematic errors due to mis-identification
• Initialize error histograms ?(Nsrc,NE).
• For each assigned event e (j,t,E),
  increment ?(i,k(E)) of the non-assigned sources i
  by ?i(e).
• In this way, we obtain the expecetd
  contamination from mis-identification at each
  energy bin.
• Neither statistical nor systematic error should
  dominate ? Choice of N !
• Total error (nik?(i,k)2)1/2

dashed statistical (Poisson) errors
NE 20 Nsrc 2
solid systematic errors from mis-identification
8
Benchmarks (1)
Object output
Implemented formulae
9
Benchmarks (2)

• Backprojection maps from segregated event
  lists of the flare of Feb 20, 2002. The (assumed)
  source locations are marked by crosses.

10
Acceptance Region at fixed N
The orbits trace out the transmissivities from
two hypothetical sources during one RHESSI
revolution.
11
Spectral inversion from counts to photons

• Standard RHESSI spectral response
• M9 M7 (M6 M5 M4 M3 M2 M1
  M0)
• Used here M M9 M7 M3 M2 M1
• The inverse M-1 is computed by SVD with condition
  number thresholding. Weak or no regularisation is
  needed for energies gt 15 keV.

Attenuator blanket
Detector resolution
Scattering in 2nd grid and atten.
Grid-pair transmission
Spacecraft scattering
Detector response to on-axis photons
Eearth albedo
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Response Matrices
(truncated SVD inverse)
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Example Flare of Nov 10, 2004
020828 020928 10 - 50 keV
Spectra based on manual source parameters (left,
solid line)
M-1
Spectral inversion
Number of accepted eventsN 1000.
14
Exploration of N
statistical error
mis-identification error
15
Comparison with OSPEX
Source 1
Source 2
Source 3
hatched total error
Errors from propagating n ? (?n ?)1/2 though
M-1
Single-Photon Method
OSPEX
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An alternative source model
CLEAN using SC 1,3,4,5
2 sources, at optimal distance (pitch / 2) for
subcollimator 5
Conservative choice!
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Jimms 1st task
Source 2 represents a diffuse background, which
can not be assigned to a source ? effect of
pile-up!
pileup?
NB the smaller source can be more frequently
excluded!
N 2000
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Jimms 1st task, ff
Source 2 (background)
Source 1
Single-Photon Method
OSPEX
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Jimms 2nd task
Reasonable source models?
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Jimms 2nd task, ff
Source 1
Source 2
Single-Photon Method
OSPEX
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Jimms 3rd task (new version)
First minute
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Jimms 3rd task (1st minute), SRM
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Jimms 3rd task (1st minute), ff
Single-Photon Method
OSPEX
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Summary

• We propose a simple method of imaging
  spectroscopy operating directly on the eventlist.
• The likelihood of origin from (known) alternative
  sources is computed, and events with ambiguous
  origin are rejected.
• The trade-off between the number and reliability
  of accepted events is a free parameter.
• The accepted events are origin-taged and
  energy-binned (for each origin class).
• The count histograms are converted into photon
  histograms using traditional spectral inversion
  substantial ambiguities are avoided by cutting
  off low energies.
• The results of the single-photon method have been
  compared to existing (OSPEX, imspec) imaging
  spectroscopy and found to generally agree.